On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
type: Item
instance of: scholarly article
title: On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
zbMATH Open document ID: 1381.37030
author: Mats Gyllenberg
author: Lei Niu
author: Ping Yan
author: Jifa Jiang
published in: Discrete and Continuous Dynamical Systems
publication date: 2017-11-14T00:00:00Z
review text: The authors study a discrete-time map \(T:\mathbb{R}^n\to\mathbb{R}^n\) where the \(i\)-th component of \(T\) is \[ T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, \] and \(0< c_i <1\); \(b_{ij},r_i>0\). They refer to this as the generalized competitive Atkinson-Allen map, which reduces to the classical map when \(r_i=1\) and \(c_i=c\) for all \(i\). The authors prove that all such maps have a simplex \(\Sigma\) which is a globally attracting invariant manifold of dimension \(n-1\). Concentrating on the \(n=3\) case they show that the dynamics on \(\Sigma\) can be grouped into 33 distinct cases, and partially analyse the dynamics in all of these cases. They find that the dynamics of the generalized competitive Atkinson-Allen model is more complex than that of the standard model.
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 37N25
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 6806961
zbMATH Keywords: discrete-time competitive model
zbMATH Keywords: Neimark-Sacker bifurcation
zbMATH Keywords: Chenciner bifurcation
zbMATH Keywords: invariant closed curve
zbMATH Keywords: heteroclinic cycle
Wikidata QID: Q62125438
reviewed by: Carlo Laing
MaRDI profile type: Publication
full work available at URL: dcds.2018027
OpenAlex ID: W2766768911
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DOI: 10.3934/DCDS.2018027
description: scientific article; zbMATH DE number 6806961